{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,8,7]],"date-time":"2024-08-07T20:40:02Z","timestamp":1723063202470},"reference-count":0,"publisher":"Wiley","issue":"33","license":[{"start":{"date-parts":[[2000,1,1]],"date-time":"2000-01-01T00:00:00Z","timestamp":946684800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2004,1]]},"abstract":"<jats:p>We consider a real semisimple Lie group <jats:italic>G<\/jats:italic> with finite center and <jats:italic>K<\/jats:italic> a maximal compact subgroup of <jats:italic>G<\/jats:italic>. We prove an <jats:italic>L<\/jats:italic><jats:sup><jats:italic>p<\/jats:italic><\/jats:sup> \u2212 <jats:italic>L<\/jats:italic><jats:sup><jats:italic>q<\/jats:italic><\/jats:sup> version of Hardy\u2032s theorem for the spherical Fourier transform on <jats:italic>G<\/jats:italic>. More precisely, let <jats:italic>a<\/jats:italic>, <jats:italic>b<\/jats:italic> be positive real numbers, 1 \u2264 <jats:italic>p<\/jats:italic>, <jats:italic>q<\/jats:italic> \u2264 <jats:italic>\u221e<\/jats:italic>, and <jats:italic>f<\/jats:italic> a <jats:italic>K<\/jats:italic>\u2010bi\u2010invariant measurable function on <jats:italic>G<\/jats:italic> such that  and  (<jats:italic>h<\/jats:italic><jats:sub><jats:italic>a<\/jats:italic><\/jats:sub> is the heat kernel on <jats:italic>G<\/jats:italic>). We establish that if <jats:italic>a<\/jats:italic><jats:italic>b<\/jats:italic> \u2265 1\/4 and <jats:italic>p<\/jats:italic> or <jats:italic>q<\/jats:italic> is finite, then <jats:italic>f<\/jats:italic> = 0 almost everywhere. If <jats:italic>a<\/jats:italic><jats:italic>b<\/jats:italic> &lt; 1\/4, we prove that for all <jats:italic>p<\/jats:italic>, <jats:italic>q<\/jats:italic>, there are infinitely many nonzero functions <jats:italic>f<\/jats:italic> and if <jats:italic>a<\/jats:italic><jats:italic>b<\/jats:italic> = 1\/4 with <jats:italic>p<\/jats:italic> = <jats:italic>q<\/jats:italic> = <jats:italic>\u221e<\/jats:italic>, we have <jats:italic>f<\/jats:italic> = const\u2009<jats:italic>h<\/jats:italic><jats:sub><jats:italic>a<\/jats:italic><\/jats:sub>.<\/jats:p>","DOI":"10.1155\/s0161171204209140","type":"journal-article","created":{"date-parts":[[2004,7,25]],"date-time":"2004-07-25T15:58:57Z","timestamp":1090771137000},"page":"1757-1769","update-policy":"http:\/\/dx.doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["An <i>L<\/i><sup><i>p<\/i><\/sup> \u2212 <i>L<\/i><sup><i>q<\/i><\/sup> version of Hardy\u2032s theorem for spherical Fourier transform on semisimple Lie groups"],"prefix":"10.1155","volume":"2004","author":[{"given":"S.","family":"Ben Farah","sequence":"first","affiliation":[]},{"given":"K.","family":"Mokni","sequence":"additional","affiliation":[]},{"given":"K.","family":"Trim\u00e8che","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2004,7,25]]},"container-title":["International Journal of Mathematics and Mathematical Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2004\/525981.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/S0161171204209140","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,7]],"date-time":"2024-08-07T19:57:35Z","timestamp":1723060655000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1155\/S0161171204209140"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,1]]},"references-count":0,"journal-issue":{"issue":"33","published-print":{"date-parts":[[2004,1]]}},"alternative-id":["10.1155\/S0161171204209140"],"URL":"https:\/\/doi.org\/10.1155\/s0161171204209140","archive":["Portico"],"relation":{},"ISSN":["0161-1712","1687-0425"],"issn-type":[{"type":"print","value":"0161-1712"},{"type":"electronic","value":"1687-0425"}],"subject":[],"published":{"date-parts":[[2004,1]]},"assertion":[{"value":"2002-09-17","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2004-07-25","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}